For the past few months, I’ve been working with David Landy on analyzing and writing up a series of number line judgment experiments. Participants in these experiments see very large numbers (e.g., 783,000,000) and respond by indicating the appropriate location on a number line with known endpoints (e.g., and ).
My focus has been on implementing a hierarchical Bayesian model of proportional judgments. At the individual subject level, the model looks (mathematically) like this:
Where is a subject’s judgment of the (proportional) magnitude of , which is the stimulus number on trial ; is an appropriately located reference point (e.g., in the model I’ve been working with, ), and indicates the smallest reference point greater than ; governs the magnitude of ‘cyclic’ responses around (midpoints between) reference points (see below); is a possibly inappropriately located reference point, expressed relative to an arbitrary fixed range; and is the number of (appropriately and/or inappropriately located) reference points.
For the data I’ve been working with, this can be simplified somewhat. is set equal to 3, and is set equal to 1. Hence, because there are only three reference points, there is only one free parameter. (In a data set that we haven’t analyzed yet, will be 4, will be set equal to 3, will be set equal to 0, and there will be two free parameters. But I digress…)
Here’s a picture illustrating predicted proportional judgments for a few different parameter values and for stimuli ranging from to :
The free parameter corresponds to the fixed reference point at . This means that if , the model predicts correct proportional judgments. (If a stimulus of were given, the first term in the model equation above would be 0. The correct proportional judgment for is 0.001, so set and solve for the free parameter, with and , and you get 1.002). The three dashed lines show predicted judgments for .
If (black lines), the model predicts judgments that are a linear function of the stimuli, while if (red lines), the model predicts overestimation of lower numbers and underestimation of higher numbers (where lower and higher are defined relative to a point halfway between the fixed reference points), and if (blue lines), the model predicts underestimation of lower numbers and overestimation of higher numbers.
If , the model predicts an ‘elbow’ in the judgment function, as if the middle fixed reference point at were thought to be halfway between and . The predicted judgments are shown by solid lines, with color indicating values as described above.
Because is so much closer to than it is to , when the stimulus numbers are expressed as in the figure above, it’s essentially impossible to see what the models predict for numbers below . Here’s the same model predictions as a function of of the stimulus numbers:
This is the basic model. In the data set we’re working on, a number of subjects give judgments that seem to be a mixture of two such models, one with a nice, proper linear value, and one with a shifted, million-in-the-middle value. So, I fit a mixture model that, for each subject, assigns each trial’s judgment to a million-in-the-middle component with probability or to a linear component with probability .
The model is also hierarchical, as noted above. This is to say that the free parameters in each individual subject’s mixture model are governed by parent distributions with parameters of their own. As it happens, one of the group-level models is a between-subjects mixture model, which allows for one group of subjects who tend toward linear responses and another who tend toward million-in-the-middle responses. I will write another post about the hierarchical structure and multiple mixtures soon.